A304209 Triangle read by rows: T(0,0)=1; T(n,k) = 2*T(n-1,k)-2*T(n-1,k-1)+T(n-1,k-2), for k = 0, 1, ..., 2*n; T(n,k)=0 for n or k < 0.

1, 2, -2, 1, 4, -8, 8, -4, 1, 8, -24, 36, -32, 18, -6, 1, 16, -64, 128, -160, 136, -80, 32, -8, 1, 32, -160, 400, -640, 720, -592, 360, -160, 50, -10, 1, 64, -384, 1152, -2240, 3120, -3264, 2624, -1632, 780, -280, 72, -12, 1, 128, -896, 3136, -7168, 11872, -15008, 14896, -11776, 7448, -3752, 1484, -448, 98, -14, 1

Offset: 0

Author

Shara Lalo, May 08 2018

Keywords

• tabf
• easy
• sign

Comments

Row n gives coefficients in expansion of (2-2*x+x^2)^n.

Examples

			Triangle begins:
   1;
   2,   -2,    1;
   4,   -8,    8,    -4,    1;
   8,  -24,   36,   -32,   18,    -6,    1;
  16,  -64,  128,  -160,  136,   -80,   32,    -8,   1;
  32, -160,  400,  -640,  720,  -592,  360,  -160,  50,  -10,  1;
  64, -384, 1152, -2240, 3120, -3264, 2624, -1632, 780, -280, 72, -12, 1;
  ...
		

References

• Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 178, 182.

Links

• Shara Lalo, Center-justified and Left-justified triangles
• Shara Lalo, Skew Diagonals in center-justified Triangle

Crossrefs

Row sums give A000012.

Programs

PARI

T(n,k) = if ((n<0) || (k<0), 0, if ((n==0) && (k==0), 1, 2*T(n-1,k)-2*T(n-1,k-1)+T(n-1,k-2)));
tabf(nn) = for (n=0, nn, for (k=0, 2*n, print1(T(n,k), ", ")); print); \\ Michel Marcus, May 10 2018